Logarithm Formula

The Logarithm is an exponent or power  to which a base must be raised to obtain a given number. Mathematically, Logarithms are expressed as, m is the Logarithm of n to the base b if bm = n, which can also be written as m = logb n. For example, 43 = 64; hence 3 is the Logarithm of 64 to base 4, or 3 = log464. Similarly, we know 103 = 1000, then 3 = log101000. Logarithms with base 10 are usually known as common or Briggsian Logarithms and are simply expressed as log n. In this article, we will discuss what is a Logarithm, Logarithms formulas, basic Logarithm formulas, change of base rule, Logarithms rules and formulas, what is Logarithm used for etc.

Logarithms Rules 

There are 7 Logarithm rules which are useful in expanding Logarithm, contracting Logarithms, and solving Logarithmic equations. The seven rules of Logarithms are discussed below:

1. Product Rule

\[\log_{b}{(P \times Q)} = \log_{b}{P} + \log_{b}{Q}\]

The Logarithm of the product is the total of the Logarithm of the factors.

2. Quotient Rule

\[\log_{b}{(\frac{P}{Q})} = \log_{b}{P}  -  \log_{b}{Q}\]

The Logarithm of the ratio of two numbers is the difference between the Logarithm of the numerator and denominator. 

3. Power Rule

\[\log_{b}{(P^{Q})} = q \times  \log_{b}{P} \]

The above property of the product rule states that the Logarithm of a positive number p to the power q is equivalent to the product of q and log of p.

4. Zero Rule  

\[\log_{b}{ (1)} = 0 \]

The Logarithm of 1 such that b greater than 0 but b≠1, equals zero. 

5. The Logarithm of a Base to a Power Rule

\[\log _{b}{b^{y}} = y \]

The Logarithm of a base to a power rule states that the Logarithm of b with a rational exponent is equal to the exponent times its Logarithm.

6. A Base to a Logarithm Power Rule 

b^logy = y

The above rule states that raising the Logarithm of a number to the base of a Logarithm is equal to the number.

7. Identity Rule

\[\log_{y}{y} = 1 \]

The argument of the Logarithm (inside the parentheses) is similar to the base. As the base is equal to the argument, y can be greater than 0 but cannot be equals to 0.

Logarithm Formulas

Below are some of the different Logarithm formulas which help to solve the Logarithm equations.

Basic Logarithm Formula

Some of the Different Basic Logarithm Formula are Given Below:

• \[\log_{b}{(m \times n)} = \log_{b}{m} + \log_{b}{m}\]

• \[\log_{b}{(\frac{m}{n})} = \log_{b}{m}  -  \log_{b}{n}\]

• \[\log_{b}{(x^{y})} = y \times  \log_{b}{x} \]

• \[\log_{b}{\sqrt[m]{n}} = \log_{b}{n^{\frac{1}{m}}} \]

• \[m \log_{b}{(x)} + n \log_{b}{(y)} = \log_{b}{(x^{m}y^{n})} \]

Addition and Subtraction

• \[\log_{b}{(m + n)} = \log_{b}{m} + \log_{b}{(1 + \frac{n}{m})} \]

• \[\log_{b}{(m - n)} = \log_{b}{m} + \log_{b}{(1 - \frac{n}{m})} \]

Change of Base Formula

In the change of base formula, we will convert the Logarithm from a given base ‘n’ to base ‘d’.

\[\log_{n}{m} = \frac{\log_{d}{m}}{\log_{d}{n}}\]

Solved Examples

1. Solve the Following:  2 log429

Solution:

Given,

 \[2 \log_{4}{29} \]

Using change of base formula n we get

\[\log_{b}{x} = \frac{\log_{d}{m}}{\log_{10}{4}}\]

\[2 \log_{4}{ 29} = \frac{\log_{10}{29}}{\log_{10}{4}}\]

\[2 \log_{4}{ 29} = 2 \times 2.43\]

= 4.86

2. Find the Value of x in log2x = 6

Solution:

The Logarithm function given above can be expressed in the exponential form as:

\[2^{6} = 64 \]

Hence, \[2^{6} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \]

3. Find log 5x + log (2x+3) = 1 +  2 log (3-x) , when x<3

Solution:

\[\log 5x + \log {(2x+3)} = 1 + 2\log {(3-x)}\]

\[\log 5x + \log {(2x+3)} = \log{10} + \log {(3-x)^{2}}\]

\[\log 5x \times (2x+3) = 10 + (3-x)^{2}\]

10\[x^{2}\]  + 15x = 10(9- 6x + \[x^{2}\])

10\[x^{2}\]  + 15x = 90- 60x + 10\[x^{2}\] \]

\[75x = 90\]

\[X = \frac {90}{75}\]

\[X = \frac {6}{5}\]

In Mathematics, Logarithm characteristics are utilised to solve Logarithm issues. Many algebraic characteristics, such as commutative, associative, and distributive, were taught to us in elementary school. There are five fundamental features of Logarithmic functions.

The Logarithmic number is connected with exponent and power, thus if xn = m, then logxm = n. As a result, we must also understand exponent law. The Logarithm of 10000 to base 10 is 4, for example, because 4 is the power to which ten must be raised to create 10000 : 104 = 10000, so log1010000 = 4.

We can represent the Logarithm of a product as a sum of Logarithms, the log of the quotient as a difference of logs, and the log of power as a product using these features.

Real number Logarithms are only seen in positive real numbers; negative and Complex numbers have Complex Logarithms.

Logarithm Applications

Logarithms have a wide range of applications both within and outside of Mathematics. Let us look at a few examples of how Logarithms are used in everyday life:

• They are used to calculate the magnitude of an earthquake.

• Logarithms are used to calculate the amount of noise in decibels, such as the sound of a bell.

• Logarithms are used in Chemistry to determine acidity or pH level.

• They are used to calculate the growth of money at a given rate of interest.

• Logarithms are commonly used to calculate the time it takes for anything to decay or develop exponentially, such as bacteria growth or radioactive decay.

• They can also be utilised in computations that need multiplication to be converted to addition or vice versa.

Instead of a simple computation, we may utilise the Logarithm table to get the Logarithm of an integer. Before calculating the Logarithm of a number, we must first understand its characteristic and mantissa parts.

Characteristic Part - The characteristic component is the entire part of a number. Any number higher than one has a positive feature, and if it is one less than the number of digits to the left of the decimal point in a given integer, it has a negative characteristic. If the number is less than one, the characteristic is negative, and the number is one greater than the number of zeros to the right of the decimal point.

Mantissa Part - The mantissa portion is the decimal part of the Logarithm number, which should always be positive. If the mantissa part has a negative value, turn it into a positive value.

How Do You Use a Log Table?

The process for determining the log value of a number using the log table is shown below. First, you must understand how to use the log table. The log table is provided as a resource for determining the values.

Step 1: Understand the Logarithm idea. Each log table may only be used with a certain basis. Log base 10 is the most often used form of Logarithm table.

Step 2: Determine the number's characteristic and mantissa parts. To get the value of log1015.27, for example, first separate the characteristic and mantissa parts.

Part of Characteristic = 15

Part of the mantissa = 27

Step 3: Make use of a shared log table. Now, utilise row 15 to verify column 2 and write the matching value. As a result, the result is 1818.

Step 4: Calculate the mean difference using the Logarithm table. Slide your finger into the mean difference column 7 and row 15, and record the associated value as 20.

Step 5: Combine the values acquired in steps 3 and 4. That equals 1818 + 20 = 1838. As a result, the value 1838 represents the mantissa part.

Step 6: Locate the distinguishing feature. Because the number is between 10 and 100 (101 and 102), the distinguishing feature should be 1.

Step 7: Finally, combine the characteristic and mantissa parts to get 1.1838.

Exemplification

Here is an example of utilising the Logarithm table to get the value of a Logarithmic function.

Determine the value of log102.872.

Solution:

Step 1: The characteristic component is 2 and the mantissa part is 872.

Step 2:  Examine rows 28 and 7 in the table. As a result, the resulting value is 4579. 

Step 3: Examine the mean difference value for row 28 and the mean difference in column 2. The value associated with the row and column is 3.

Step 4: Adding the numbers from steps 2 and 3, we get 4582. This is the mantissa section.

Step 5: Because the number of digits to the left of the decimal part is one, the characteristic part is less than one. As a result, the characteristic portion is 0

Step 6: Finally, join the characteristic and mantissa parts. As a result, it becomes 0.4582.

As a result, log 2.872 = 0.4582

Quiz Time

1. Which of the Following Statements is Not True?

1. \[\log_{7}{7}\] = 1

2. \[\log ( 4 + 3) = \log {( 4 \times 3)} \]

3. \[\log_{10}{1} = 0 \]

4. \[\log ( 4 + 5 + 6) = \log{4} + \log{5} + \log{6} \]

2. What is the Value of \[\frac{\log{\sqrt{8}}}{\log{8}} \]?

1. \[\frac{1}{\sqrt{8}}\]

2. \[\frac{1}{4}\]

3. \[\frac{1}{2}\]

4. \[\frac{1}{8}\]